The spikes a bright star wears

Almost every photograph of a bright star taken through a large telescope shows the star with four sharp spikes, a cross of light far longer than the star’s disc. The spikes are so familiar that artists draw stars with them, and so constant that they are easy to mistake for a property of the stars. The first images from the James Webb Space Telescope showed stars with eight — six long ones and two shorter — and some pictures taken with small refracting telescopes show none at all.

The spikes are the telescope’s, and specifically they are the thin vanes, or “spider”, that hold its secondary mirror in front of the primary. The connection between a set of thin metal strips inside a tube and a cross of light on the sky is diffraction, and it is exact: the image of a point of light formed by a telescope is the squared Fourier transform of the telescope’s aperture, the pattern of where light is let through. The image that is a diffraction pattern twice found that a lens takes the light from an object apart into its spatial frequencies and puts it back together; for a single star the object has every frequency equally, and what is left in the image is the aperture’s own transform. Read the aperture as a pattern, transform it, and the star’s appearance follows — spikes included.

An aperture, and its picture of a star #

A clear round aperture of diameter $D$ images a star as the Airy pattern: a central disc of angular radius $1.22\lambda/D$ and a series of fainter rings, which how far apart two things have to be found set the resolution of every instrument. A reflecting telescope’s aperture is not clear: the secondary mirror sits in the middle of it, blocking a central disc, and that obstruction moves light from the central disc into the rings, as the rings that belong to the edge found, since rings are the transform of the aperture’s edges and an obstruction adds an edge.

The secondary has to be held, and it is held by vanes running from the tube’s wall to the mirror’s mount. Each vane is a thin straight strip across the aperture, and a thin straight strip has a very particular transform. Along its length it is long, so its transform in that direction is narrow; across its width it is thin, so its transform in that direction is wide. A thin vane’s transform is a streak — narrow along the vane’s own direction, long at right angles to it.

Every panel in that figure is a computation, not a picture of a real telescope: a pupil sampled on a grid of 512 by 512 points, with the vanes’ edges anti-aliased, transformed in two dimensions and squared. Two checks are built into it. The brightness at the centre must be the square of the open area, since the zero-frequency term of a transform is the sum of what is transformed; and the total light in the image must equal the light through the pupil, Parseval’s theorem. Both hold to the precision of the arithmetic.

The same rule without the transform #

The Fourier transform states the answer; the wavelets that Huygens’ construction uses explain it. Treat every point of the open aperture as a source of wavelets, all in step, and ask in which directions they arrive at a distant screen together. Along the axis, every wavelet travels the same distance and they all agree: the bright core. Off the axis, wavelets from different parts of the aperture travel different distances, and the brightness depends on how those differences are distributed.

A vane takes away a strip of sources. Seen from a direction perpendicular to the vane’s length, every point along the strip is at the same distance; the missing sources would all have arrived in step, and removing them takes away a contribution that was coherent along the whole strip — so in that plane the change to the image is large. Seen from a direction along the vane’s length, the missing sources are spread over many path differences and their contributions would have cancelled among themselves; removing them changes almost nothing. The vane’s effect is concentrated in the plane at right angles to it, which is where the spike lies.

It is the single slit again, in the version where rays stop being enough began with: a slit narrow in one direction spreads light in that direction and not in the other, because its narrowness is what makes the path differences across it small. A vane is a slit’s complement — an opaque strip where the slit was a transparent one — and by Babinet’s principle it diffracts the same pattern with the opposite sign. A thin vane spreads its light widely across itself, the long way of the spike, and not at all along itself.

Counting spikes #

Each vane gives two spikes, pointing in opposite directions, because the transform of a real pattern is symmetric through the centre. A cross of four vanes is two straight bars, each running right across the aperture, and each bar’s two spikes lie along the same line as the opposite vane’s — so four vanes make four spikes, two pairs. Three vanes at 120° are not opposite one another. Each throws its two spikes at right angles to itself, and no other vane’s spikes coincide with them, so three vanes make six spikes.

The rule makes the number of spikes a fingerprint of the telescope. Four spikes mean an even number of straight vanes arranged in opposite pairs, the most common design because it is simple and stiff. Six mean three vanes, a design that blocks less light but is less rigid. A star with no spikes was photographed through a refractor, which has no secondary, or through a reflector whose vanes are not straight.

Vanes are not the only straight edges in a telescope. The Webb telescope’s primary mirror is a mosaic of eighteen hexagonal segments, and its outline is a hexagon whose three pairs of straight edges each throw two spikes — six in all. The segments’ internal gaps are straight too, and run in the same three directions, reinforcing them. The secondary is held by three struts, two of which run along directions that coincide with the hexagon’s spikes; the third, running vertically, throws a pair of horizontal spikes of its own. Six from the shape, two from the strut: the eight-pointed star is a portrait of the mirror’s outline and one strut, read through its Fourier transform.

Why a spike reaches so far #

What makes the spikes conspicuous is not how bright they are near the star but how slowly they fade. Between the spikes the light is the round aperture’s ring pattern, whose brightness falls on average as the inverse cube of the angle from the star. Along a spike it falls as the inverse square.

The difference is a difference between edges. A curved edge, such as the round rim of the aperture, sends its diffracted light in every direction — at any angle only a short stretch of the rim is oriented to contribute — and the wave that comes from the rim found that only the parts of an edge perpendicular to the direction of observation send light there strongly. A straight edge is perpendicular to one direction along its entire length, and in that direction every part of it contributes together. The spike is the direction in which the whole vane adds up. At 20 times $\lambda/D$ the spike is 24 times brighter than the light between spikes at the same distance, and the ratio grows further out.

That is why spikes dominate long exposures. A telescope photographing a field in which one star is a thousand times brighter than the faint galaxies around it records the bright star’s core saturated, its rings merged into a halo, and its spikes running across the frame far beyond the halo, because the spikes fade more slowly than everything else the star’s light makes. The spikes are as much a property of the photograph’s exposure as of the telescope: they are always there, and a longer exposure reaches further down their slow decline.

Light borrowed from the star #

A spike is not free light. The vanes block part of the aperture, and the light that would have passed through that part is missing from the image. By Babinet’s principle — the light diffracted by an obstacle is the light that an aperture of the same shape would let through, with the sign reversed — the vanes’ own diffraction pattern is exactly what is subtracted from the clear aperture’s image, and that pattern is the spikes. The spikes carry the light the vanes block, spread along their length.

The two curves trade against each other unequally. The star’s peak, which sets how faint a star can be detected against the sky, falls as the square of the open area, so vanes covering a few per cent of the aperture cost the peak only a few per cent. The light thrown into the spikes grows in proportion to the vanes’ area, and it is put precisely where it does most harm: in narrow streaks far from the star, where faint objects are looked for. Making the vanes thinner gains almost nothing in peak brightness and reduces the spikes in proportion. Telescope designers make vanes as thin as their stiffness allows, and mount them edge-on to the incoming light, which is why a secondary mirror is often held by blades a millimetre thick and tens of centimetres deep.

The obstruction of the secondary itself does something similar to the rings. How accurate a mirror has to be used the peak’s brightness, the Strehl ratio, as the measure of an image’s quality; a central obstruction lowers the peak and moves light into the first ring, which is why planetary observers, who look at fine detail near the centre of the image, prefer telescopes with small secondaries or none.

Curved vanes, and where the light goes instead #

If straight edges concentrate their diffracted light into spikes because every part of them faces the same way, an edge that faces every way should spread its light evenly. That is the principle of curved vanes, used in some telescopes built for photographing faint objects.

The curved vanes block nearly the same area as the straight ones, so by Babinet they divert nearly the same light from the core. They do not remove the light; they redistribute it. A curved vane faces every direction along its length, so at any angle only a short stretch of it contributes, as for the round rim, and its diffracted light fades as the rim’s does, as the inverse cube. A bright star then has no spikes, only a slightly brighter halo, which is a better background for finding faint things near it — at the cost of a halo in every direction instead of streaks in a few.

The same reasoning designs the masks used to find faint planets beside bright stars. A coronagraph’s aperture is shaped so that its transform has a region, on one side of the star, where almost no light falls at all, and the edges are chosen — curved, tapered, or with their transmission graded smoothly — so that their diffraction goes elsewhere. The dark hole such a mask makes is the opposite of a spike: a direction in which the pupil’s edges have been arranged to cancel instead of to add.

Starbursts in cameras and eyes #

Telescope vanes are not the only straight edges light passes. A camera lens stopped down to a small aperture forms that aperture with a ring of straight-edged blades, the iris diaphragm, and the aperture is a polygon. A polygon with an even number of sides throws as many spikes as it has sides, its opposite edges pairing up as the bars of a cross do; one with an odd number throws twice as many, its edges having no opposite partners. That is why a night photograph of street lamps taken at a small aperture shows each lamp as a starburst, and why a photographer can tell from the number of points how many blades the lens has: fourteen points from a seven-bladed diaphragm, eight from an eight-bladed one.

The eye makes starbursts of its own. A bright point seen at night through a dilated pupil is often surrounded by fine radial streaks, and they come from straight or nearly straight structures in the eye’s optics — the junctions at which the fibres of the crystalline lens meet, and the edges of the eyelids and lashes when the eye is partly closed — each spreading light at right angles to itself as a vane does. The pattern is personal: no two eyes have quite the same lens structure, so no two people see quite the same star.

A focusing tool made of spikes #

Spikes also have a use. A Bahtinov mask, a sheet with three groups of parallel slots at slightly different angles placed over a telescope’s aperture, produces three sets of spikes from a bright star. When the telescope is exactly in focus, the middle spike passes exactly between the other two; when it is out of focus, the middle spike shifts to one side, and its displacement is far more visible than the blurring of the star itself. Amateur astronomers focus their cameras this way to a precision of a few micrometres. It works because a slotted pattern is a set of straight edges whose transform is a set of spikes, and because defocus shifts the different groups’ spikes by different amounts — what a thousand slits buy found that a grating’s diffracted light goes in sharp directions fixed by its period, and a Bahtinov mask is three gratings at different angles, used to make the image’s focus a matter of where three lines cross.

What the figures leave out #

The figures compute the diffraction pattern of a pupil in a single colour, for a perfect optical system with no aberrations and no atmosphere. A real star’s image is formed in a band of colours, and since the spikes and rings scale with wavelength, the outer parts of a spike are smeared into a short spectrum — the spikes in colour photographs show it as coloured fringes along their length. Atmospheric turbulence blurs the core and rings into a seeing disc for ground-based telescopes, but the spikes, made by straight edges extending over the whole aperture, survive it far out from the star. The vanes are drawn as perfectly straight, opaque, thin strips; real vanes are blades seen edge-on, with some thickness along the light path, and slight bends or misalignments split a spike into two close ones. The pixelation of the computation leaves faint grid artefacts in the outermost parts of each panel. The domain is monochromatic, aberration-free imaging of a point source by a pupil of known shape.

Still open: how dark the space beside a star can be made #

The search for planets around other stars needs images in which a star’s light is suppressed by ten thousand million times within a few $\lambda/D$ of it, and every edge in the telescope’s pupil — the vanes, the segment gaps, the outline — scatters light into exactly the region where planets are sought. How to design pupils and masks that reach that contrast on segmented telescopes with struts, whether the edges can be hidden by shaping the aperture or must be compensated with deformable mirrors that sculpt the light’s phase, and how close to the theoretical limit a real instrument can get given the mirror’s imperfections, are the central problems of planet imaging, and the instruments planned for the next space telescopes are built around them.

The spikes themselves are understood exactly. A telescope’s image of a star is the squared Fourier transform of its pupil, and each straight vane across the pupil throws two spikes at right angles to itself — four from a cross, six from three vanes at 120°, six more from a hexagonal outline, eight on Webb’s images — whose light falls as the inverse square of the angle against the inverse cube of the rings, 24 times brighter at 20 λ/D; by Babinet they carry the light the vanes block, and curved vanes spread the same light into a glow. A star’s spikes are a drawing of the telescope that took the picture.